Medial Axis Transform - Digital Geometry in Image Processing - page 208

(a) (b)

FIGURE 6.13: Normal computation at the contour points for the image
circle for B={112} (a) dense normal map, (b) sparse normal map.
Reprinted from Pattern Recognition Letters , 23(2002), J. Mukherjee et al., Use of Medial Axis Transforms for Com-
puting Normals at Boundary Points, 1649-1656, Copyright (2002), with permission from Elsevier.

in Fig. 6.13 by reproducing it from [153]. In Fig. 6.13(a), the normals computed
at the contour points of the circle using digital distances for B = {112}
are shown. In this figure the lengths of the normals are proportional to the
magnitudes of N p 's as computed by the algorithm NCUM. Also, for the clarity
of the presentation, only sparse normal maps are shown in Fig. 6.13(b). The
value of Nthresh was kept as 1 in this case.
Typical examples of computed normals for some of the objects of known
geometry using different octagonal distances are also shown in Figs. 6.14, 6.15,
and 6.16. In these figures, only results with good distances such as {112},
{1112}, and {12} are shown.

6.5.3 Quality of Computation
In [153], a quantitative measure is used for judging the quality of the nor-
mal computation. In this measure, analytical maps of normals at the boundary
points of objects of known geometry are used. For example, in Fig. 6.17 (a),
(b), and (c), analytical normal maps for circle, square, and rectangle, respec-
tively, are shown. Let n p be the unit normal vector at a point p ∈C obtained
from the algorithm NCUM. Let the analytical value of unit normal vector at
a point p ∈ C be denoted as m p (Fig. 6.18). Then an error measure E n is
defined as

E n = 1− |n p · m p | /| C |

∀p∈C

where | C | is the number of points in C.
The value of E n becomes zero when n p and m p lie along the same direction.
The larger the deviation, the greater is the contribution in the aggregated error
measure (E n ). However, if the algorithm NCUM fails to compute n p at any